Valuation of Boundary-Linked Assets by Stochastic Boundary … · 2016-12-07 · be applied to...

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An Intemational Joumal Available online at www.sciencedirectcom computers &

. c , . . c . (~o , . . cT . mathematics with applicationl

Computers and Mathematics with Applications 52 (2006) 137-160 www.elsevier.com/locate/camwa

Valuation of Boundary-Linked Assets by

Stochastic Boundary Value Problems Solved with a

Wavelet-Collocation Algori thm

M. E S T E B A N - B R A V O A N D J . M . V I D A L - S A N Z * D e p a r t m e n t of Business Admin i s t r a t i on

Univers i ty Carlos III of Madr id CL. Madrid, 126. 28903 Getafe, Madrid , Spa in

j vidal@emp, uc3m. es

A b s t r a c t - - T h i s article studies the valuation of boundary-linked assets and their derivatives in continuous-time markets. Valuing boundary-linked assets requires the solution of a stochastic differential equation with boundary conditions, which, often, is not Markovian. We propose a wavelet- collocation algorithm for solving a Milstein approximation to the stochastic boundary problem. Its convergence properties are studied. Furthermore, we value boundary-linked derivatives using Malli- avin calculus and Monte Carlo methods. We apply these ideas to value European call options of boundary-linked assets. (~ 2006 Elsevier Ltd. All rights reserved.

K e y w o r d s - - S t o c h a s t i c boundary value problems, Financial derivatives, Wavelets, Collocation methods.

1. I N T R O D U C T I O N

In this paper we consider pricing boundary-linked assets and their derivatives in continuous-time markets. The values of these assets are contractually linked at several dates by means of boundary constraints. Therefore, valuing boundary-linked assets requires the solution of boundary value stochastic differential equations.

A stochastic boundary value problem (BVP) is defined as

dZt = b (t, Z t ) dt + a (t, Z t ) dWt, for t E [0, T], (1)

with a boundary condition

~ ( x ) = c, (2)

where Wt is a d-dimensional Brownian motion, Xt a continuous time d-dimensional stochastic process, ~ a continuous linear operator from the trajectories' space to R d, and e E R d constant.

*Author to whom all correspondence should be addressed. We thank Professor A. Balbas and Professor E. Galperin for their helpful comments and suggestions. This research has been supported by two Marie Curie Fellowships of the European Community programme IHP under contract numbers HPMF-CT-2000-00781 and HPMF-CT-2000-00449, respectively.

0898-1221/06/$ - see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by A A/~-TEX doi:10.1016/j.camwa.2006.08.010

138 M. ESTEBAN-BRAvO AND J. M. VIDAL-SANZ

For example, a boundary condition can be a terminal condition with a(X) = X T , o ~ ( X ) =

AoXo "b ATXT where A1 and AT are real matrices, or a more involved condition such as a (X) = f T d A t X t , where At is a d x d matrix whose components are functions of bounded variation in [0, T]. The theory of stochastic BVPs has also considered some cases of nonlinear operators a, see, e.g., [1]. Other references on boundary value stochastic differential equations are [2,3] and [4].

Stochastic BVPs typically arise from the application of the Pontryagin's maximum principle to stochastic control optimum problems with finite time horizon T, where the boundary condition is given by the transversality condition (see, e.g., [5, Proposition 10.1, pp. 112-113]). These systems cannot usually be analytically solved, and algorithmic tools are required to cope with these problems. Despite recent contributions in stochastic BVP literature (see, e.g., [6] and [7], that focus on Stratonovich integrals), much can be done to enlarge the catalogue of techniques

for solving BVP. In this paper, we propose a projection-based method for solving stochastic BVPs. Its main idea

consists of using a wavelet-collocation method to solve a finite-difference Milstein approximation to the stochastic differential problem. We prove that this procedure provides a strong approximation for the solution to (1) and (2). We study the numerical performance of the algorithm in

several examples. We apply these ideas to study the valuation of boundary-linked assets and their derivatives.

The analysis of boundary-linked assets is not only a theoretical problem, but can also be applied to the increasingly exotic assets traded in actual economies. With the growing sophistication of financial markets, investors are demanding new, more complex options products, tailored to their needs. In particular, there is an increasing number of financial assets whose values are contractually linked at certain periods of time, such as leases and rental agreements. An illustrative example is the English real estate lease market. In English law, two legal estates exist in buildings and land: freehold (absolute ownership which does not expire) and leasehold (temporal possession for a specified time period). Leasehold enables liability on covenants to pass from tenant to tenant and indeed from landlord to landlord. In this context, the lessor bears the risk associated with the residual market of the asset at the maturity date of the contract, and the buyer bears the short-term lease risk, where the value fluctuation of the lease randomly fluctuates subject to some boundary constraints, e.g., a zero value of the leasing contract at the maturity date. The value of lease assets can be formulated by a second-order boundary value

stochastic differential equation

J((t) = b(t)X(t) + W(t), t e [0, T], (3)

X(O) = p, X (T ) = 0; (4)

that is, the acceleration of lease assets prices is proportional to their growth rate and affected by a white noise shock. Note that any second-order problem can be reduced to a first-order system of stochastic differential equations with boundary value conditions in the space of states, see, e.g., [4]. Hence, to value boundary-linked assets, we are faced with the problem of solving stochastic differential equations with boundary conditions.

Often, the solution of stochastic BVPs does not satisfy the Markovian conditional !ndependence property, see, e.g., [8] and [9]. Therefore, standard Black-Scholes arguments cannot sometimes be applied to value derivatives of boundary-linked assets. We propose the use of Malliavin calculus to value these derivatives. In particular, we consider the generalized Clark-Ocone formula and present a procedure for its computation based on the Monte Carlo method and wavelets approximations. To illustrate this methodology, we consider a European call option of boundary- linked assets.

The rest of the paper is organized as follows. After some preliminaries, Section 2 provides a brief introduction to stochastic BVPs. In Section 3 we present an algorithm for solving boundary value stochastic differential problems. Its numerical performance is illustrated by means of some

Valuation of Boundary-Linked Assets 139

examples. Next we study the properties of the solution approximation. In Section 4 we apply these ideas to value boundaxy-linked assets, of which prices are determined by a stochastic differential equation with boundary conditions. Also, we consider the valuation of boundary-linked derivatives and their numerical computation. Finally, proofs axe placed in Appendix A and a MATLAB code is included in Appendix B.

2. S T O C H A S T I C B V P S : P R E L I M I N A R I E S

Now we introduce some basic notation and tools that will be used through the paper.

White Noise Process

Let S be the Schwartz space in IR and let S ' be its dual (the space of tempered distribu- tions) endowed with the weak-* topology and Borel subsets B. By the Minlos theorem, there is a probability measure tt on ,9' such that fs, exp{i(w,¢}}d#(w) = exp{-(1/2)ll¢l122(m)} for

all ¢ • S, where (w, ¢) is the evaluation of w • £ ' on ¢. The space (8', B, #) is the white noise probability measure, satisfying the It6 isometry E~[(w,¢)] -- IICH22(R) for all ¢ • S. Consider d independent realizations from #, then we construct a d-dimensional Wiener process Wt = ((Wl, I[0,t]),..., (Wd, I[0,t]))', which has a continuous modification in C(R; R d) with Wt = 0 for t _< O, and such that (w, ¢) = fR ¢(t) dWt for all ¢ • S, in the sense of ItS's integral. Consider the Gelfand triple S c L2(#) C 8 ~, where

L2(#) = = 2 (5)

An orthogonal basis for L2(#) is given by the family {Hk}, indexed by all vectors k = (kl, • . . , k~), • ., = m h with {kj}~= 1 C N, for all m = 1,2,. where Hk(w) 1-Ij=l kj((w, ej)) and {h~}, {e~} are

Hermite polynomials and Hermite functions, respectively. Then, we define a singular white noise generalized process as follows:

) ) ' ~/t ~ ~-~I , • • • , COd , (6)

with I;Vt(w) = ~;'~k ek(t)Hk(w). A detailed review of this topic can be found, e.g., in [10] and [11].

Let ~ = C0([0,T]; ]R d) be the space of all the continuous functions in [0,T] which vanish at zero, with T > 0 deterministic. The restriction of the Wiener process Wt to [0, iv] induces a Borel probability space, whose completion is denoted by (~, .4, P). Let {At} denote the filtration generated by Wt, completed and made right continuous.

Malliavin Calcu lus :

Next we introduce some tools from Malliavin calculus. For all h • L2([0,T]; Rd), consider

W(h) = f [ h8 dWs. Let C ~ ( R ~) be the set of functions f : R n ~ R infinitely continuously differentiable such that f and all its derivatives are bounded. We denote by 7:) the set of real random variables of the form F -~ f ( W ( h l ) , . . . , W(hh)), with f • C°~(R~) for any n • N and h l , . . . , h n • L2([O,T];Rd). For all F • T) we can define the differential operator DF as the stochastic process

D

DtF = > ~. ~--:-f (W ( h i ) , . . , W (hh)) hi(t), Oxj Vt C [0, T], (7)


and the i terated differential D~ ..... t F = Dr, . . . Dt, F, and D°F = F. Let D q,p be the closure of 79 with respect to the Sobolev norm

l[ )--; {,,...,tf :,::o,*lJl ) n p

I[F[[q,p = F[Ip + D Lp(n) j=l Lp(n)

= E[IF[Pl+~-~E Dr: ..... ,¢F d h . , \ j= : ,T]J

(s)

with p • (0, c~), and D q'°~ as the elements F in D q,2 with finite norm

n

IIFIIq'~ = lIFIIL•(n) + -= L~[[0,T]J] L=(n) (9)

We also define D °~,p = Nq>: Dq'P and D °° = Nq,p>: Dq,p.

The operator D : D :'2 C L2(fi) --* L2([0,T] x ~) is known as the Malliavin derivative of F 6 D 1'2, with first-order derivatives and second-order moments. The adjoint operator of D,

denoted by 5, is defined for all processes u such tha t E[f[0,T ] DtFut dr] <_ cllFIIL2(fl ). If u 6

Dora(5), then E[5(u)F] = E[f[o,T ] DtFudt] for all F • D 1,2. Often, the operator 5 is expressed

as 6(u) = fo T ut dWt, and is known as the Skorohod integral. I t can be proved tha t 6(u) is equal to the It6 integral if the process ut is adapted. The duali ty can be used to establish the Clark- Ocone formula, see, e.g., [12]; that is for all F • D 1'2, F -- E[F] + f [ E[DtF [ .At] dWt. For an introduction to the Malliavin calculus and its properties, see, e.g., [13-15] and [16]. Malliavin derivatives can be also considered as Fr4chet derivatives, see, e.g., [16]. MaUiavin calculus can

also be introduced using Wiener-It6 chaos expansions, see, e.g., [17].

S t o c h a s t i c B V P S o l u t i o n s

Consider the stochastic BVP (1) and (2). Although the s tudy of existence of solutions for these problems is beyond the scope of this paper, we will sketch a proof using a scheme similar to that of the determinist ic case. Notice that there exists a unique solution associated to D~c(t) = 0 with a(x) --- c since a are linearly independent (at least over Ker{D}). Consider a Green's matr ix of functions G(t, s), such tha t any g 6 C0([0, T]; R d) with Dg integrable can be expressed as follows:

T g(t) = Po(g)(t) + fo G(t, s)Dg(s)ds, (1o)

where Po(g) is the unique element in Ker{D} which agrees with a(g) . Furthermore,

T Dg(t) = DtPo(g)(t) + fo DiG(t, s)g(s) ds. : (11)

Let ~1,2 = {h 6 L2([0,T] x f~;R d) : 30 E (D:'2) a, Do = h}. We will express the stochastic BVP (1) and (2) in a more convenient way, using the following property: define Dx = u, thus

u = G[x] and G-:[u] = x, reciprocally, with

T

G[x](t) = Po(x)(t) + fo G(t, s)x(s)ds,

G-:[u](t) = Dt {Po(x)(t)} + fo T DiG(t, s)x(s) ds.

(12)

(13)


Therefore, defining U = G(X), and the nonlinear operator

T[U](t) := b (t, G-I[UI(t)) + a (t, a - l [Ul ( t ) ) l;Vt, (14)

we can express the stochastic B V P as U = T[U]. Then, we can guarantee the existence of solution in B V P by proving the existence of a fixed point U ° for T, for which it suffices that T is a continuous retraction in a space isometric to 7-/1,2, and a pathwise unique solution U ° exists. Also, X ° --- G - I ( U °) is the almost sure (a.s.) unique solution of BVP, with Ut ° = G(V°)( t ) the Malliavin derivative of X °. Since

[/o ] E [tlT[U]II2L=[O,TI] = E b ( t ,a - l [U]( t ) ) + a ( t ,O- l[Ul( t ) ) Wt 2 dt

= [llb( ,a-lI<(t))ll + < (15)

T is contractive if IIb(t,x)H <_ kllxll , Ilcr(t,x)ll < cllxH, for all t, x, and (k + c)HG-1[[ < 1. Note that the solution X ° satisfies

]o t x g = x ° + b °) d , + ,, ( s ,X° , ) e W , , (16)

for some initial condition Xo °. Given X °, we can define a sequence {X~} as follows:

f[ f0 t X ~ +1 -- X ° + b (s, X~) ds + a (s, X'~) dWs, (17)

where E [ f [ IX~ -X°t I2dt] ---, 0 under appropriate Lipschitz conditions on b and ~. Hence, using (17), the existence of a unique continuous version of X ° on [0,T] can be proved using arguments similar to the case of ordinary stochastic differential equations.

Sufficient conditions for fixed points can be restrictive and if no solution exists, a weaker concept can be used by generalizing the notion of fixed point. For a 5 > 0, a point U0 is called a &point of T if IIT(Uo) - U0H < 5 and a set of &points is called a &set (see [18, p. 1553 D. Therefore we can consider a set Xo = G-I(Uo) of &equivalent solutions which may exist even when the exact solution does not.

Adaptativeness

The solutions of stochastic BVPs are anticipative in nature due to the boundary condition. But given an appropriate (anticipating) initial condition, the dynamics of the process is driven by an ordinary stochastic differential equation. This is the logic underlying shooting numerical methods. These methods are widely used to solve deterministic BVP (see [19] for a review), and have been recently extended by [6] to solve Stratonovich stochastic BVPs. A shoot!ng method is a successive substitution method based on the idea of guessing the initial condition until its associate solution satisfies the boundary condition.

We use the shooting argument to define a conditional adaptativeness for solutions of BVPs. Let A0 be the completion of the smallest a-algebra such that {or(X°)} is measurable, and consider the filtration {Set}t>0 with 5rt = .40 n .At. Note that conditioning on { a ( X °) = c}, the unique solution X ° satisfies (16), where X ° is .40 measurable, and as a consequence X ° is adapted respect to {~-t}t>0. Therefore, conditioning on the boundary condition expression (1) can be considered as an It6 integral or, alternatively, a generalized process (see [11 D. Otherwise, equation (1) should be interpreted in terms of Skorohod stochastic integrals.

142 M. ESTEBAN-Bm~vo AND J. M. VIDAL-SAsz

3. A N A L G O R I T H M T O S O L V E S T O C H A S T I C B V P S

The numerical resolution of stochastic BVPs is the aim of this section. We propose a wavelet projection-based algorithm for solving stochastic differential equations with boundary conditions. Its main idea consists of using a wavelet-collocation method to solve a finite-difference approximation to the stochastic BVP. With this end in view, we first introduce some concepts of wavelet approximation.

Within the last decades, wavelet multiresolution methods have proved to be a flexible method for approximating relatively irregular functions with a parsimonious number of parameters. The first wavelet basis can be at least traced to the [20] work, but the theoretical foundations of wavelets have been established by physicists and mathematicians i~om the early 1930s to the 1980s. The interest on wavelets has increased since [21] and [22] introduced the use of multiresolution as a framework to study wavelets expansions. A historical perspective can be found in [23] and [24]. Excellent monographs in wavelets are [22-25] and [26].

Given the Hilbert space L2(~), let consider a sequence of closed subspaces {Vn}n~z such that:

(i) V~ c v~+l, Vn ~ Z, (ii) Anez V~ = {0}, and

(iii) U~ez v~ is dense in L2(R).

In particular we say that {V~}~ez is a multiresolution if each subspace V~ is the span of an orthonormal basis {¢n,k}keZ, with ¢,~,k(t) = 2n/2¢(2'~t - k), and ¢ E L2(R) is known as the father wavelet. This concept was introduced by [21].

As {¢mk}keZ are orthonormal, if IIv,(x) denotes the orthogonal projection of an arbitrary x G L2(R) into V,~, then

x(t) = lim Hv~(x)(t) = lim E (x'¢mk)L, Cmk(t), (18) n - - - ~ o o n - - ~ o o

kEZ

in the sense of L2. Whenever ¢ has compact support, for each t E R the summation in (18) contains a finite number of nonnuU terms. Otherwise it should be truncated for practical applications. In practice, one of the most popular wavelets systems is the compactly supported wavelet proposed by Danbechies, see [23].

Due to the fact V0 C I/1, any father wavelet can be expressed as

¢( t ) = a k ¢ ( 2 t - k), (19) kEZ

for some {ak}keZ G 12. Taking Fourier transforms, we can express (19) as

where ~ is the Fourier transformed of ¢ and A(w) = (1/2) ~-~keZ ake-ik~" A multiresolution can be defined by finding a function A(w), which means finding a sequence {ak}, such that the Fourier inverse of ~ satisfying (20) is a father wavelet. Daubechies proposed a procedure to construct a finite sequence {ak} so that

2N-1 ¢(t) = E ak¢(2t-- k) (21)

k=l

and the resulting ¢ has compact support and 2N - 1 vanishing moments (in other words, N de- termines the number of nonzero coefficients in (18) and is an index of the Daubechies wavelets).

r 12N--1 Furthermore, the coefficients ~ak$k=l can be computed recursively. For a detailed exposition about Daubechies wavelets of order N and their properties, see [23].

Let W~(R) be the Sobolev space of functions (a.s. identical) with L2-integrable weak derivatives up to order r. If x E W~(N), under appropriate conditions, wavelets derivatives can also


approximate the weak derivatives of x. The multiresolution ideas can be specialized to the space L2([0, T]) taking a multiresolution {V,~}~>0. In this context, it can be proved that IIy~(x) ~ x uniformly for all x E C([0,T]), see, e.g., [27].

The first basic step of our algorithm is to consider a real wavelet multiresolution { n}n=l in L2([0, T]). To simplify notation throughout the remainder of the paper, given a vector of d functions x(t) = ( x l ( t ) , . . . , xd(t)) ' , we will denote the wavelet approximation of any x E L2(R) d by

nv~ (x)(t) = ~ o~,~¢.,k(t), (22) kEZ

where 0n,k E ]~d is a vector of coefficients. Henceforth we will consider compactly supported wavelets such as the ones proposed by Daubechies; otherwise we should truncate the summation in (22).

The next step is to consider a finite-difference approximation to the stochastic differential equation. For the sake of simplicity, we first consider the problem of solving an autonomous stochastic system dXt = b (Xt )d t + o ( X t ) d W t , with a ( Z ) = c. In particular, we consider Milstein's [28] finite-difference approach,

Xn (ti,n) -- Xn (ti-l,n) : h~b (X~ (t,-1,n) ) + o (Xn (ti-l,,~)) (Wt,,~ - Wt,_~,~)

00 (X~ (t i- l ,n)) dWs~ dns2 + o (x~ (t,_~,~)) ~ -~ ° ,

X,~(0) = c. (24)

The double stochastic integral can be readily computed, e.g., in the scalar case

= ~ (~(w~,~ - ~,_~.) -

For the multivariate case see [29, Chapter 5, Section 8]. Thus, the third and final step of the algorithm consists of applying the wavelet-collocation

method to the Milstein approximation and solving the following system of nonlinear equations in O,~,k E R:

E 0n,k (¢n,k (ti,n) - Cn,k (t i- l ,~)) = h~b (X~ ( t i - l ,~)) kEZ

+ U o~,~,~(t,_~,.) ~ o~,~.,~(t,_~,~) ( ( w ~ , , ° - ~ ~ - h ~ ) , (26) \ kEZ

0~,k~ (¢~,k) = c, (27) kCZ

0* where ti,,~ -- 2-hi C [0, T] with i C Z, and hn = 2 -n. The solution coefficients { k,n} determine X ~ e V ~ a s

Xo*,n(t) = E 0:, k~)n,k(t)' (28) kEZ

where only a finite number of terms are different from zero. Often system of equations (26) and (27) is nonlinear and has to be solved by numerical methods.

There are numerous methods for solving nonlinear equations (see, e.g., [30]). However, we consider Newton's method for systems of nonlinear equations (see [31, pp. 86-92]) because of its rapid

144 M. ESTEBAS-BRAvO AND J. M. VIDAL-SANZ

rate of convergence for solving large systems of smooth nonlinear equations. Denote by F(8) = 0 the system of equations (26) and (27). In essence, the Newton algorithm consists of computing a search direction At~ so that VF(~t)At~ = -F(8~), and updating t~z+l = ~z + A~ iteratively from a starting guess t~0.

Unfortunately Newton's method only ensures local convergence. In the (possible) presence of multiple local optima, an obvious probabilistic global search procedure is to use a local algorithm starting from several points distributed over the whole optimization region. But this procedure is extremely inefficient as when many starting points are used, the same minimum will eventually be determined several times. In the last decades the development of mathematical programming theory has led to significant advances in global optimization related modelling, algorithms, and real-world applications. In particular, global optimization procedures have been considered in the literature which could be used for solving system of equations (26) and (27) such as the delta algorithm [32, Chapter 10]. Alternatively, an implementation of the cubic algorithm in MAPLE to compute global optima can be found in [33].

Equation systems derived from collocation methods are often ill-posed (although in Examples I and II, the condition number of the Jacobian matrix V F at the numerical solution is moderate). In the case of ill-conditioned problem Tikhonov's [34] regularization method can be considered, which is equivalent to consider the Levenberg-Marquardt method (see [31, p. 227]) instead of the Newton algorithm.

Solving BVPs with a(t) = a for all t is particularly easy. In this case, Milstein's equations are reduced to the Euler-Maruyama approximation (see [35]) and the system of equations (26) and (27) in 0,~,k E R to be solved is

X,~ (t~,,~) - X,~ (t i- l , ,~) = (ti,,~ - t i - l , ,~) b ( X n (t¢-l, ,~)) + (Wt~,. - W t , _ , , . ) , (29)

(X~) = c, (30)

where Xn(ti,,~) = ~keZOn,k¢,~,k(t~,n), ti,n = 2-hi C [0, T] with i C Z, and hn = 2 -'~. Also, this method can be applied to the nonantonomous stochastic systems, dXt ~- b(t, X t ) dt+

a(t, X t ) dWt, with a (X) = c. However, instead of the Milstein equation, we should consider an expansion for nonhomogeneous stochastic differential equations, see, e.g., [29, Chapter 5, Section 5].

In case of using 5-point notion, we have a b-set of coefficients {0~,n} which can be computed by the spherical algorithm proposed by [18].

In order to illustrate the accuracy of the method, we compute several examples of stochastic BVPs with analytical solution and use the compactly supported wavelets of Daubechies. The algorithm has been implemented and the tests have been carried out on MATLAB 6.5 on an Intel Centrino Pentium M 1.6GHz with machine precision 10 -16. First we consider a very simple stochastic BVP to show how to set up parameters to compute its approximate solution. In Appendix B we present the MATLAB code to clarify exposition of the proposed method.

EXAMPLE I. Consider the problem

dXt = dWt, t c [0,1], (31)

X1/2 + X1 = O. (32)

This problem has a solution of the form X ( t ) = - (1 /2 ) (W1/2 + W1) + Wt. We compute the numerical approximation of its solution for a sample path of {Wt} using

Daubechies wavelets with N = 3 and the step length h = 2 - 2 (i.e., n = 2 and the number of dyadic points used is 9). To compute the solution path of this problem, we consider the Milstein approximation and we solve the system of equations (26) and (27) in 0n,k E R (in this case, it is a nonhomogeneous linear system). Direct methods (usually variants of Gaussian elimination)


are implemented in the core of MATLAB and are made as efficient as possible for general classes of matrices.

In order to illustrate the accuracy of the numerical solution, we compute the solution path for 400 realizations of the Brownian motion {Wt}. The average approximation error IIX(t~,n) - Xe. ,n( t i , ,OIl~ = max{t,.~} IX(t~,n) - Xo,,~(t~,,~)l is 2.7717 x 10 -15 and its s tandard deviation 1.5896 x 10 -15.

In case of being interested in a higher accuracy, we can consider a larger number n. For n = 6, the condition number of the Jacobian matrix V F at the numerical solution is 93.4161, which means that we can expect to lose approximately two digits of precision during the computation of {0~,n}. Figure 1 shows the values of the actual solution and its approximation over the dyadic points ti = 2-6i E [0, 2] with i C Z.

Eu~r Al~rox.

O.

-0,

-1

-2

0 ~.x <_.2

Figure 1. Numerical and exact solution of Example I with N = 3 and n = 6.

The next example is intended to demonstrate tha t the algorithm also behaves well in more complicated problems. However, a higher number of dyadic points (in other words, higher pa- rameter n) should be considered to get accuracy.

EXAMPLE II . Consider the problem

dXt = dWt , t e [0, 1], (33)

~0 1 X t = (34) 0.

The solution of this problem has the general form X ( t ) = - f~ Wt dt + Wt . For a given sample path of {Wt}, using Danbechies wavelets with N = 3, Figure 2 shows the

exact and the computed approximation for n = 2, 4, 6.

Table I reports the approximation errors to the solution and the computational cost for solving the stochastic BVP for n = 2, 4, 6. In this case, the approximation solution with n = 6 gives the smallest residual l L X ( t ~ , n ) - Xe.,n(t~,.)ll~o as illustrated in Figure 2, for which the condition number of V F at the numerical solution is 620.8568, which means that we can expect to lose approximately three digits of precision during the computational process.

Similarly to stochastic BVPs, most stochastic differential equations arising in real-world applications cannot be solved exactly. Numerical methods to get accurate solutions are Euler- Maruyama and Milstein schemes, among others. A review of the literature can be found, e.g., [29]. The proposed algorithm can also be used to solve ordinary stochastic differential equations as the following example illustrates.

- 0

-0.6

-0.~

F o r n ~ 2 .

M . E S T E B A N - B R A v O A N D J . M . V I D A L - S A N Z

l i i i i i L i i 0.1 0.2 o.a 0.4 O,S 0.6 0.7 0,$ O.J

O _ < x < 2

I I °3 o:, o.., oi, o:, oi,, o:*

O ~ x < 2

F o r n = 4.

146

1

0.8

0.6

0.4

O.2

4 ~

~ . 4

o:, o~

0.6

0 . 4

0.2

- 0

- - 0 , 4

-0 .6

-,0.8 0.1 0.2 0.3 0 4 O.S 0 6 0 7 0.8 0.9

O~.x<_.2

For n ~- 6.

Figure 2. For n ---- 2, 4, 6, numerical and exact so lut ions of E x a m p l e II w i th N = 3.

Table 1. Approx imat ion errors and running t imes for comput ing E x a m p l e 2 w i th

n----2

n = 4

n = 6

n = 2,5,6.

[Ix (t,,~) - x e . , ~ (**,~)ll~ c P , (Seconds)

0.2058 0.02

0.0997 0.03

0.0075 4.66

EXAMPLE I I I . Consider the problem

dXt = bXt dt + aXt dWt, t E [0, 1], (35)

with X0 -- 4, and W0 -- 0 a.e. The solution of this problem has the general form

X(t) = (exp ( ( b - a---~) t + aWt) . (36)

Assume tha t b = 2, a = 1, and ~ -- 1. We compute the numerical Solution of this problem using Daubechies wavelets with N = 3. Figure 3 shows tha t the approximat ion error is sat isfactory for n -- 6, a l though there is room for improvement in the r ight-hand side of the t ime interval and the ill-posedness of V F (as i ts condit ion number is 30559).

- - , - s o ~ , ~ ' Numerical Aprax.


. 0'., 0h °'.s 0'., 0'.s 0'0 0'., 0'~ 0'., O<x<_.2

Figure 3. Numer ica l a nd exact solutions of E x a m p l e I I I wi th N = 3 an d n = 6.

Although we have focused on solving stochastic BVPs with linear boundary conditions, we can also apply this method to problems in which the boundary conditions are given by nonlinear continuous operators. When a is a nonlinear continuous operator, the proposed method can be applied by replacing equation (27) by a ( ~ k e z On,k¢~,k) = c. However, the convergence theory for this type of problem is beyond the scope of this paper.

3.1. Convergence Analysis

In this section, we study the convergence properties of the proposed method. Proofs are placed in Appendix A. Assume:

A.1. Let {V,~} be a multiresolution in L2(St), with compactly supported father wavelet ¢, and assume for all x E W~(]R), with 1 < r < q, q >_ 1, and all integer v, 0 _< I[vlll < r - 1, it is satisfied

[[D ~x - D~Hv. (x)iJL 2 = O (2-(r-il~lh)~) . (37)

Whenever x E C ~ (St) with compact support the same rates are satisfied replacing the L2 norm by the supremum norm. In spaces L2([a, b]), an analogous behavior is assumed.

There are several sufficient conditions for this result that can be found in the literature, often based on the regularity of order q assumption. The father wavelet ¢ is said to be regular of order q E N, if ¢ has a version q times continuously differentiable and for 0 _< IIulll < q, and any positive integer p e N, there exists a constant C v > 0 such that IDle(t)] < (1 + IItN)-pcp, y t e R. See [22] for further details.

A.2. Let X°( t ) be a solution of the stochastic BVP and define C = { ( t ,X°( t ) ' ) ' : t E [0,T]}. Also, assume that b,a E C2(A f) where Af C R R+I is an e-neighborhood of C in the Loo norm, and for some ~ > 0, it is satisfied

Pr~Lte[0,T]inf d e t ( I - D = b ( t , X ° ( t ) ) - D , a ( t , X ° ( t ) ) l ~ V t ) > 7 } = 1. (38)

Notice tha t the last condition is satisfied whenever det ( I - D~b(t, x) - D~a(t, x)g(t)) is nonnull for all (t ,x) and for all g E LI[0,T]. For example, when a(t) does not depend on X it suffices d e t ( I - D=b(t,x)) ¢ 0 for all (t ,x). In particular, for linear stochastic BVP, b(t,x) --- bx and a(t, x) = a, it suffices that det(I - b) ¢ 0.


We start with an auxiliary result on the rate of approximation of the wavelet-Galerkin method. Given the multiresolution {V.}, let x . E V,~ be the wavelet-Galerkin solution to the stochastic BVP; i.e., x . satisfies

Hv. { D x . - b ( t , x . ) - a ( t , x . ) I ; V t } = O , a ( x . ) = c. (39)

THEOREM 1. Let consider the problem B V P with solution X°( t ) , and a multiresolution sequence {Vn} in L2([0,T]). Assume that A.I , A.2 are satisfied, then there exist 5 > 0 and an integer M such that X ° is unique a.s. in B ( X °, 6) = {X : IIX -X°[ Ioo <__ 6}, and the projected system

1-Iv, { D X , - b ( t , X , ) - a ( t , X , ) l;Vt} =O (40)

has an a.s. unique solution X , E Vn N B ( X °, 6). Fttrthermore, with probability one,

max {IIx,~ - xolloo , I I D x , - Dx°llo~} = 0 ( 2 - " ) . (41)

In order to prove the convergence of the wavelet-collocation method we will use an interpolation result.

THEOREM 2. Consider a multiresolution {V,} in L2(N) satisfying A.1. For each x E L2(R) with an a/most everywhere (a.e.) continuous version with compact support, we define I 'v . (x) as any function gn 6 Vn such that gn(t.,i) = x(tn,i), for all {t.,~ = 2-ni}iez. Then, there exists a unique element in Pv. (x). Furthermore, assuming:

1. ¢ is regular of order q _> 1, and 2. the Poisson summa ~-~kez ~(w + 27rk) > 0, for almost every w E [0, 27r], being ~(w) =

fR ¢( t)eit~ dt the Fourier transformed ore;

for all x E wq(N) with compact support, there exist K > 0 and no such that V n > no,

I I rvo (x ) - ~IIL2 < K I l n v . (~) - ~llw~ - (42)

Given the multiresolution {V,}, let x , E Vn denote the wavelet-collocation solution to the stochastic BVP, and therefore

I 'v . { D x , - b ( t , x , ) - a ( t , x , ) l ; V t } = O , a ( x , ) = c . (43)

The following result is a consequence of Theorems 2 and 1.

COROLLARY 3. Under the assumptions of Theorems i and 2, the wavelet-collocation method satisfies the approximation property at rate O(2-n).

Therefore, it remains to prove the consistency of the proposed method based on the Milstein scheme

~ ( ( W t i , ~ - t i-l , .) , (44)

for 2 . e y . .

PROPOSITION 4. Under the assumptions of Theorems 1 and 2, let 5c~ E Ira be the approximation generated by the proposed method and xn the solution of the wavelet-collocation method. Then, it is satisfied II~- - x . l l ~ = G ( 2 - " ) .


4. B O U N D A R Y - L I N K E D F I N A N C I A L M A R K E T S

Consider a monetary bond and d boundary-linked assets. Let us assume that the bond has a continuous positive price per share Xo(t) solving the stochastic differential equation

dXo(t) = r(t)Xo(t) dr, Xo(0) = 1, (45)

where r(t) is a progressively measurable process satisfying f0 T Ir(t)l dt < oo. Therefore Xo(t) = exp{f o r(s)ds} for t • [0, T]. Let X~(t) denote the price per share of the ith boundary-linked asset and X(t) = (Xl ( t ) , . . . ,Xd(t))'. Assume that the initial values of the boundary-linked assets XI (0 ) , . . . , X d(0) are positive constants almost surely. For each t • [0,T], suppose also that these prices are governed by

dX(t) = b (t, X(t)) dt + a(t) dW(t) (46)

and the boundary conditions fl(X) : p, where fl(X) is a set of d linear continuous real functionals and p • •d. A particularly relevant example is the linear boundary value stochastic differential equations defined as

dX(t) = b(t)X(t) dt + a(t) dW(t), (47)

with fl(X) -- p. Following the arguments given in [9], this problem possesses a unique solution in CI([0,T]) if and only if det{fl(xS)} # 0 for some s • [0,T] (equivalently for all s • [0 ,~) , where xS(t) is the solution of the homogeneous system dx(t) : x(t)b(t) dt with x(s) : Id; i.e., problem dx(t) : x(t)b(t)dt with ~(x) : 0 has only the trivial solution. In this case, we can express

X(t) = d-1 (t)p + G(t, x)a(s) dW(s), (48)

with Green function

[r[]oSJ_i(u)v(du)_ l[o,s](t)Id] ] d(s), (49) G(t,s) J - l ( t )

l[o,8](t) being the characteristic function of the set [0, s]. In the linear context, we define a portfolio (80 (t), 0z ( t ) , . . . , Od(t))' as a progressively measurable

process that represents the number of units of the assets for each t • [0, T]. The value of a portfolio is given by Ve(t) = Oo(t)Xo(t) + Oo(t)'Z(t). The portfolio Oo(t), O(t) : (01(t),. . . , ~d(t)) ! is called self-financing (with respect to Xo(t), X(t)) if

fo T (s)00(s)'Xo(s) + O(s)'b(s) + 10(s)' k(s)l 2 & < k = l

dV6(t) = t?o(t) dXo(t) + O(t)' dX(t).

(5o)

(51)

Then, Vo(t) = Vo(O) + fd Oo(s) dXo(s) + ]~ O(s)' dX(s). Notice that given an appropriate O(t)', there exists Oo(t) such that (Oo(t), O(t)') is self-financing. A self-financing portfolio (Oo(t), O(t)')' is called admissible if its corresponding process Vo(t) is a.e. lower bounded; i.e., 3 Ko > 0 such that Vo(t) >_ - K o a.e. for all t • [0,T]. This is a natural constraint in real life as debts cannot infinitely increase. An admissible portfolio is called an arbitrage in the considered market, if the associated value process satisfies Vo(O) and Vo(T) > 0 a.e. with P(Vo(T) > O) > O.

In practice, the value of a portfolio is often discounted at the nonrisk rate. This means that we can normalize prices by defining J~0(t) = 1 and f~k(t) = Xo(t)- lXk(t) . Given a self- financing portfolio, the discounted values are ~ (t) = Lo (t)-1Vo (t), and applying the It5 formula dVo(t) = O(t)'dL(t). Therefore, the self-financing portfolio property is not affected by the dis- count normalization. Furthermore, if (Oo(t), O(t)') is admissible for (L0, L(t)), then (Oo(t), O(t)') is also admissible for the normalized market (1, L(t)), as r(t) is bounded.

150 M. ESTEBAN-BP, AVO AND J. M. VIDAL-SANZ

4.1. V a l u a t i o n of B o u n d a r y - L i n k e d Der iva t i ve s

In this section we consider pricing of boundary-linked derivatives. In this context, standard Black-Scholes techniques cannot help to value derivatives of boundary-linked assets as these processes are not Markovian. However, an alternative approach based on the generalized Clark- Ocone formula can be considered. We illustrate this approach considering a European call option of boundary-linked assets.

Let XT be the values of the d boundary-linked assets at the maturity date of the contract. By the Clark-Ocone formula,

// XT(W) = E[X] + E [DtXT(w) I ~t] dWt, (52)

with Dt XT(W) is the Malliavin derivative of XT(W). The Clark-Ocone formula can be extended to study ~-T random variables G(w) that are stochastic integrals respect to processes

T 17Vt(w) = fo a(s, ~) ds + Wt(w), (53)

where a(s, w) is an ~t adapted stochastic process satisfying some appropriate regularity conditions. By the Girsanov's theorem, 14zt is a Wiener process under certain probability measure Q on ~'T, where dQ(w) = ZT(W)dP(w), with

Zt(w) = exp - a(s ,w) ds - a(s ,w) 2 ds . (54)

The generalization of the Clark-Ocone formula ensures that if G(w) is a regular stochastic integral respect to l~t, then

# G(w) = EQ [G] + ~aO. (t, w) dlTVt, (55)

where ~o(t , w) = EQ[D,G- afT Did(s , w) dITVs I ~t]. The proof and other technical details can be found in [16].

This result can be applied to the valuation of derivatives in linear boundary-linked markets. As the value of a portfolio is given by Vo(t) = 8o(t)Xo(t) + O(t)'X(t), we have

Oo(t) = X0(t) -1 (Vo(t) - O(t)'X (t) ) . (56)

If the portfolio is self-financing,

dVo(t) = Oo(t) dXo(t) + O(t)' dX(t) , (57)

using (45), (47), and (56), we obtain that

dVo(t) = (r(t)Vo(t) + (b(t) - r(t)) O(t)'X(t)) dt + O(t)'cr(t) dW(t) . (58)

Assuming that the solution of the boundary value problem is ~', adapted, our aim is to find a portfolio O(t) leading to the lower bounded ~T measurable random variable G(w), such that G(w) = Vo(T) and the initial value is Vo(0).

If Vo(t) is ~-t adapted, taking a(t) = (b(t) - r(t))a(t) -1 and ITVt = f [ a(s) ds 4- Wt, we can express

dVo(t) = r(t) Vo(t) dt 4- O(t)' a(t) diTV(t). (59)

Therefore, the discounted portfolio Vo (t) = Xo (t)- 1Vo (t) satisfies

df/o(t) = Xo(t)- lO(t)t a(t) dlTV(t). (60)


By the generalized Clark-Ocone theorem, the discounted final value

G : : ~do(T) : Xo(T)- !Ve(T) (61)

verifies

As a consequence, I~0 (0) = EQ [G] and the required portfolio is

(62)

(63)

This expression can be applied to the analysis of derivative prices in boundary-linked markets in an analogous way to the Black-Scholes formula.

For example, consider a European call option which gives the owner the right to buy the stock with value XT at exercise price p. Then, G --- (XT -- p)+ represents the payoff at time T. Clearly, G -- fp(XT) where fp(x) = ( x - p ) + . Note that fp(.) is continuous but not differentiable at x = p and DtG cannot be obtained applying the chain rule. However, fp E C([0,T]) can be approximated by a sequence {fn} C CI([0, T]) with f~(x) = fp(x) for ] x - p [ > l /n, and 0 < f~ < 1. Taking Gn = f~(XT) we have

D,G : limoo DtG,~ : I[p,oc)(XT) . Dt XT : I[p,oo)(XT) . XT" a(t). (64)

Hence,

(65)

In particular, if dX(t) -: bX(t) dt + a dW(t) and r(t) = r > 0, then Dta = Dr(b-~')O " - 1 = 0 a.e., and

O(t) = Xo(t)EQ [Ib:,,o~) (XT) XT I .T't] • (66)

For any t < T, the value of the derivative at time t is determined as the value Vo(t) of the portfolio O( t ).

When XT follows a diffusion process, (66) leads to the classical Black-Scholes formula applying Markovian arguments. However, in case of boundary-linked assets markets, as these assets follow a boundary value stochastic differential equation, the expectation in (66) cannot be computed using Markovian arguments and numerical resolution methods are required.

In order to compute the portfolio {Ot}, associated to a given a realization of the underlying processes {(Xt, Wt)), we propose the use of a Monte Carlo simulation-based estimation of (66) using independently generated realizations of the process {Xt} conditioned to the information set ~'t, which is computed using the wavelet-collocation approach presented in Section 3. Three steps are involved. For each dyadic point ti,n ---- 2-'~i C [0,T] with i E Z, we simulate M independent realizations of the Brownian motion, denoted by W]~,~ for all j = 1,. ; . , M, such

that W~ -- Wt, for any dyadic point t E [0, ti,,~]. The second step of the algorithm consists of solving, for each j -- 1 , . . . , M, the following system of boundary value stochastic differential equations:

dx{ = bx{ dt + odw';, (67)

( x J ) = p, (6s)

152 M. ESTEBAN-BRAVO AND J. M. VIDAL-SANZ

with the addi t ional constraints X~ = Xt, for any dyadic point t E [0, ti,,~]. In part icular , we compute the solution of these problems J M {Xt }j=t by means of the wavelet-collocat ion algori thm presented in Section 3. In the th i rd and final step, we compute portfolio (66) at each ti,n e [0, T] as

1 M 8 M (ti,,~)= exp (rt i , ,~)~ ~ I[p,oo) (X~) X~. (69)

j = l

For example, Figure 4 shows the numerical s imulat ions of Xt condit ioned to the available

information at ti,~ = 0.25, when Xt follows the boundary-va lued s tochast ic differential equation given in Example I and M = 50. Figure 5 shows the pa th of portfolio (66) computed as (69) with r = 0.2 and p -- 0.25.

20

15

10

5

0

-5

-10

-15

-20

Monte Cado of X(t)IF(t=.25) I '

011 " I & o13 oI, 0.'5 ols o, & o; O<_x<_l

Figure 4. Monte Carlo simulations of Xt I .~t=0.25.

2

1 . 8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

I + Computed ponfo4io I '

/ /

/

0:1 °2 °.3 0 , 0s & 01, & 01, O < x < l

\ \

Figure 5. Path of portfolio (66), with r = 0.2 and p = 0.25.


A P P E N D I X A

P R O O F S

(A) P r o o f o f T h e o r e m 1

We will use the following theorem.

THEOREM 5. Let B be a Banach space, {Vn} c B a sequence of increasing linear subspaces, and IIv. a sequence of continuous projections converging pointwise to the identity operator on B. Let T define a (nonlinear) operator in B. I f (1 - T)u = 0 has a solution u °, T is continuously Frdchet differentiable at u ° and (1 - T'~o)u = 0 has only the trivial solution in B, then u ° is unique in some sphere B(u° ,5) = {u E B : I[u - u°[[ <_ 5} for some 5 > O, and there exists an integer M such that for all n > M the equation Hv~{(1 - T)u} = 0 has a unique solution u,~ 6 V~ N B(u °, 5). Moreover, 3 K > 0 such that

II - oll < ,¢ o - °11 (70)

PROOF. See, e.g., [36, Theorem 5]). |

Using the properties of the Green function and the continuity of b, the functional T is continuous relative to the uniform norm on a neighborhood of u ° = G(x°). Since for each realization of the white noise process (1 - T ) u -- 0 can be seen as an equation in Co([O,T],Nd), we will consider the equation Hv~ (I - T)u~ = 0 in Vn.

First, we check the continuous Fr4chet differentiability of T. For any u E B ( u °, 5) define h = u - u °. Notice tha t Af contains all line segments in R R+I such as {u ° + Oh : 0 6 [0,1]}, since

T x(t) - x°(t) = / o DtO(t , s)h(s) ds, (71)

with IIx - x°llL~ < ~ whenever 5 is small enough, using that

X := esssup/Tr [DtG(t, s)[ ds < c~, (72) te[0,T] J0

where esssup denotes the essential supremum with respect to the Lebesgue measure (i.e., esssup f is the smallest number a C R for which f only exceeds a on a set of measure zero).

Recall tha t u ° = G(x°). We will see tha t the Fr4chet derivative T at u°(t) = Dx°( t ) respect to the direction h = (u - u °) is given by

and the error term is given by

~o T euo(u)(t) = IIIh[ll 2 (1 O)b" ( t ,u°( t ) + Oh(t)) dt

+ IHhtll 2 /oT(1 -- O)a" ( t ,u°( t ) + Oh(t)) dW( t ) ,

(73)

(74)

b", a" being the second directional derivatives of b(t, .), a(t, .), respectively, in the direction h/lllhll h and II[hlll 2 R ---- >-~r=l Ilhrl] 2' Clearly Ile~o(u)llLo¢ <_ Clllu--uOII2L~, where cl is the maximum between X and sup{b"(t, z) + a't(t, x ) W T } over all directions on cl(Af), the closure of Af, which is finite with probability one as Pr(WT = c~) = 0 for finite T.

Notice also that T~0 (h) (t) can be expressed in the original domain as the operator

T~o(x) = (D~b ( t ,x°( t ) ) + D~cr ( t ,x° ( t ) ) IiVt) Dx. (75)


Since d e t { ( I - D~b(t, x° ( t ) ) - D ~ a ( t , x°(t))l;Vt)} ¢ O, almost surely, for all t E [0, T], there exists a unique trivial solution for

( I - D~b ( t , x ° ( t ) ) - D~a ( t , x ° ( t ) ) l;Vt) D x = O, (76)

with a(x) = c. This implies the same result for ( I - T~o)U = 0, and hence assumptions of Theorem 5 are satisfied.

Thus, there exists an integer M > 0 such that, for all n > M, a solution u~ E V. exists and is unique in the same sphere. Moreover, there exists a constant c > 0 such that un = D x m u ° = D x °, and

I1,,,~-,,°ILL_ -< c Ilrlvo ~ ' ° - u° l lL . • (7;)

By the Banach-Steinhaus theorem, for all u E Vn,

IIHvJ - ~°11~ = IIH~,, (~o _ ~) _ (~_ ~°)11~ = 11( I - fly.)(u ° - ~)IIL~

_< c'inf { 1 1 ~ ° - - % : ~ ~ vo} = o ( 2 - ~ ) , (78)

where the rate 0 (2 -~) follows from Assumption A.1. The result follows noticing that [[Dx~ - Dx°[[L~ = ][un - u°[[L~, and

(79)

using that x~ - x ° = G - l ( u n - u°).

(B) P r o o f o f T h e o r e m 2

Theprob lem of interpolation in Vn at points t~,i = 2-'~i can be reduced to solve the problem go(i) = x(t~,~) in g0 C V0 and then take gn(t) = 2~/2go(2nt). Therefore, assume that go(t) = ~ k e z Ok¢(t - k) solves this problem, i.e.,

ok¢(i - k) = x ( t . ,d . (80) kEZ

Clearly, a unique solution exists since {¢ ( t -k )}ke z are linearly independent functions. To simplify the notation, we denote xi -- x(t,~,i), and hence ~ k e z O k ¢ ( i -- k) = xi. This i s a convolution equation that we will solve in the spectral domain. Let us define the discrete Fourier transform of C b y

~(~) = ~ ¢(k)e -~k~. (81) kEZ

The Poisson formula states tha t ~)(w) -- ~ k e z (I)(w + 2~rk). If ¢ is regular of at least order 1,

this series converges uniformly on compact sets. Furthermore, as (~(w) > 0 a.e. for w E [0, 2~r], the inverse has a Fourier expansion (1/~(w)) = ~ k e z f l k e -ikw where b := ~ k e z [flkl < ~ , by the Wiener-L6vy theorem. Thus, we can explicitly evaluate the coefficients {0k} as

0k = ~ / ~ k - , x , . (82) iEZ

Obviously,

~-~ ~k-ixi 2 I = b 2 E [ x ( t n , i ) [ 2 , (83) iEz

with supn>l ~ i e z ]x(t'~,i)[ 2 < ~ as x is continuous with compact support.


Next, we will prove tha t

IIr-o (x)IIL~ -< bllxll-, (84)

where I[xll, = (2 - n ~ i e z [x(tn#)12) 1/2, applying Schwartz's arguments as in [37].

Notice tha t IIg, llL2 = 2-"llgOIIL2 = 2-~II~(g0)HL2, where ~'(go)(w) is the continuous Fourier transformed of g0. We will prove tha t II~'(g0)ll2 = ~keT. ]0k] 2 and the result follows. Let us define ~(w) = ~ k s z 9k e-~k~, then

( / (~)2 fR ~(~) ( / ~ IIJ:(go)ll~: = ~ J: y~OkCo,k d~ = :)-~Oke -'k~ d~

\ k E Z / \ k E Z /

2~ ~ + 27rk) 2 = fo 15(w)12 ~(w dw

(85)

27r

kEZ

as {¢(t - k)}kez is or thonormal if and only if Y~kez I~( w + 21rk)l 2 = 1 a.e., for details see [23]. Hence, we have tha t NFv.,(x)H~2 < b2]lxll2n .

Defining x,~ = Hv~(x), we have tha t Fv~(xn) = xn since x,~ e V~ and has compact support. And as a consequence,

IIr.o(~) - ~11~ = IIrvo (~. - ~) + ~ . - ~11~ -< b2 I1~- - ~11. + I1~. - ~IIL~

= b 2 Ilnvo (x) - xll. + IInv~(x) - ~IIL~ • (86)

Moreover, as for all x E W~(R), with r > 1,

/f } Ilxll 2 < C I. ¢ - 2 ~ 13r(x)(w)l 2 dw + 2-'~]]xll~v~ , (87)

see [38], the result follows applying the same bound to I ] n y , , ( x ) - ~II~,

(C) P r o o f o f Proposition 4

Let X n be the solution to (43), and )(n be the solution to (44), i.e., the proposed algorithm. We will prove tha t E[IIX ~ - X,~II~] = O(h2) and the result follows.

Consider first the autonomous case. By assumption b, a c C2(Af). Define the operators,

O 1 02 b L ° - - - - L 1 (88)

= b-~x -~ 2 Ox 2' = a-~x"

By the Wagner and Platen expansion (see [29], for a review), Xn satisfies the equations

Ai,~ (X~) = Ri,~ (X~) , (89)


for all i E Z such that ti,,~ = 2-hi E [0,T], where

gi , , ( Z , ) = Z~ (t~,n) - X~ (t i - l , , ) - h,~b (X~ (ti-,,~)) - a (X~ (t~-x,,)) (Wt,,, - Wt,_l,.)

- - L l a (Xn (ti-l,n)) ~-'1 dWz dWs, (90) , - - 1 , ~

Ri,,~ (Xn) = LOb (Xn(z)) dz ds + Lib (X,,(z)) dWz ds , t~'-- 1, n J t i - - 1 n --1 , n

+ L°a (Xn(z)) dz dWs t i - l ,~ - 1 , n

f t " , ,

g* s z

~- L1Lla (Xn(u)) dWu dWz dWs. , - -1 ,n - 1 , 4

Let A,~(Xn) = l~(Xn) denote this system of nonlinear equations, where E[HRn(Xn)II 2] = O(h2). On the other hand, 2n satisfies system (44); i.e., An(f(n) = 0. Then,

where D A ~ is the Fr@chet derivative at some intermediate point ~ . Since HDA~ (.)]]~ > e > 0 uniformly, it is satisfied that

and the result follows. For nonautonomous systems the argument is analogous.

A P P E N D I X B

M A T L A B C O D E F O R E X A M P L E I

The proposed algorithm can be efficiently implemented using MATLAB 6.5 (http:// '~w. ma~hworks, corn). The wavelet toolbox in MATLAB provides a comprehensive collection of routines for applying wavelet approximations. We present the code for solving Example I, the simplest example, because it provides enough information to understand how construct a code for other problems.

The first two steps of the implementation are: the computation of the Wiener process and the Daubechies wavelets. A Wiener process is the sum of zero mean Gaussian variables, computed with the help of remda(1,dimension) , an internal MATLAB function that produces a 1-by-dimension matrix with random entries, chosen from a normal distribution with mean zero, variance one, and standard deviation one (we also call r a n d n ( ' s z a Z e ' , 100) to reset the genera- tor to the state 100). To generate Daubechies wavelet, we use wavefun from MATLAB'S wavelet toolbox. These wavelets have no explicit expression. MMlat's cascade algorithm gives a con- structive and efficient recipe for defining compactly supported wavelets such as Daubechies (for details see [23]). By typing wave in fo ( ' db ' ) , at the MATLAB command prompt, you can obtain a survey of the main properties of this family.

The next step consists of solving the system of equations (26) and (27) in 0~,k C • where b(t, Xt) = O, a(t, X,) = 1, a(Xt) = X1/2 + X1, and c = 0. The specific algorithm used for


solving this system depends upon the s t ructure of the coefficient ma t r ix A. For this example, we use the backslash opera tor to solve simultaneous l inear equations. In general, the opt imizat ion toolbox can solve systems of nonlinear equations (see the reference page in the online MATLAB documenta t ion for more information). % EXAMPLE I: dx(t)=dW(t), x(.5)+x(1)=O, t in [TI,T2] ?. Solution x(t)=-.5*(W(.S)+W(1))+W(t) %%%%%%%%%%%%%%%%%%%%%%%%%?.%%%%%%?.%%%%%%%%%%%%%% % P a r a m e t e r s mu=O ; sigma=l ; TI=O; T2=IO; %%%%%%?.%%%%%%%%%%%%%%%%%%?.%%%%%%%%%%%%%%%%%%%%% ?.First step: Definition wavelet to approximate N=3; ?. N, Daubechies' order

n=4; ?. n, resolution level wname=' db3' ; % wavelet name [phi,psi,x] = WAVEFUN(wname,n) ; step = x(2)-x(1); ?.size of approximation step dimension=length(x) ;

h=2" (-n) ; n_vars=T2*2^n- i- (TI*2"n+2-2*N) +l ; ?. number of

c o e f f i c i e n t s t h e t a s ?. D e f i n i t i o n of Wiener p r o c e s s r andn ( ' s t a t e ' , 100) dW= s q r t ( h ) * r a n d n ( 1 , d i m e n s i o n ) ; ?. Wiener i n c r e m e n t s W = cumsum(dW); % d i s c r e t i z e d Wiener p a t h W= [O;W'] ; ?. Analytical solution exact_x= -. 5* (W(. S*2^n+l) +W(l*2^n+l) ) +W; ?.?.?.?.?.?.?.?.?.•%?.?.••%?.?.?.?.?.•?.•?.?.?.•%?.?.?.%?.?.?.?.?.?.?.?.?.?.%?.?.?. %Second step: Computation of K=F(thetas), system of equations (26) and (27)

?. First : Equation (26) for i=Tl*2^n+2-2*N : T2*2~n-i, inner_prodl ( :, i- (TI*2"n+2-2*N) +I) =zeros (dimension-l, l) ;

for I=i :dimension-i, phi_il (i) =0 ; if (((2"n)*x(l+l)-i>=O) • (((2^n))*x(l+l)-i<=2*N-l)), phi_if(1) = (2^(n/2)) * phi(((2^n)*x(l+l)-i)*2^n+l);

end end inner_prodl ( : , i- (TI*2"n+2-2*N) +i) = phi_il ( : ) ;

inner_prod2 ( :, i- (TI*2"n+2-2*N) +i) =zeros (dimension-l, i) ;

for i=i :dimension, phi_J2 (i) =0 ; if (((2^n)*x(1)-i>=O) ~ (((2"n))*x(1)-i<=2*N-l)), phi_J2(1) = (2"(n/2)) * phi(((2"n)*x(1)-i)*2"n+l); end end inner_prod2 ( :, i- (Tl*2^n+2-2*N) +i) = -phi_J2 (i : dimension-1) ' ;

end R=inner_pr odl+inner _pr od2; ?. Second: Computation of last equation of system (27), in this case x(.5)+x(1)--C

aux_Kl=zeros (n_vars, i) ; aux_K2=zeros (n_vars, I) ;

158 M. ESTEBAN-BRAVO AND J.M. VIDA~SANZ

f o r j=TI*2^n+2-2*N:T2*2"n-1 , i f ( ( ( 2 " n ) * . 5 - j > = O ) ~ ( ( 2 ~ n ) * . 5 - j < = 2 , N - 1 ) ) ,

a u x - K l ( j - ( T l * 2 ^ n + 2 - 2 * N ) + l ) = ( 2 " ( n / 2 ) ) * p h i ( ( ( 2 ^ n ) * . 5 - j ) * 2 ^ n + l ) ; end end f o r j=TI*2"n+2-2 ,N :T2*2"n -1 ,

i f ( ( ( 2^n ) - j>=O) ~ ( ( 2 ^ n ) - j < = 2 , N - 1 ) ) , a u x - K 2 ( j - ( T l * 2 ^ n + 2 - 2 * N ) + l ) = ( 2 ^ ( n / 2 ) ) * p h i ( ( ( 2 " n ) - j ) * 2 ^ n + l ) ; end end K=[R;aux_Kl '+aux_K2'] ; Z The r i g h t s i d e of sys tem (27) b = [ d W ( l : d i m e n s i o n - 1 ) ' ; O ] ; % Th i rd : Solve Kx=b. Note t h a t i n t h i s example i t i s l i n e a r . The b a c k s l a s h o p e r a t o r employs d i f f e r e n t a l g o r i t h m s to h a n d l e d i f f e r e n t k i n d s of c o e f f i c i e n t m a t r i c e s . theta=K\b; ZXZXXZZZ~XX~X~XX~ZZ~X~XXXX

Informative results error_sisz=K*theta-b;

Approximation error norm(error_sist,inf)

Condition number of K. cn=cond(K)

Digits of precision that we can expect to lose during the process d=log(cn)/log(10) XZ~XXXZZZXXX~XX~X~XX~XXX~XXZXXX

Thi rd s t e p : To compute t he n u m e r i c a l s o l u t i o n g i v e n t h e o p t i m a l c o e f f i c i e n t s t h e t a s .

x = O : 2 " ( - n ) : 2 ; a p r o x = z e r o s ( l e n g t h ( x ) , l ) ; f o r k = Tl*2^n+2-2*N:T2*2"n-1 , for i=l:length(x), phijk(i,k-(T1*2^n+2-2*N)+l)=0; if (((2^n)*x(i)-k>=0) a (((2~n)).x(i)-k<=2*N-1)),

phijk(i,k-(Tl*2"n+2-2,N)+l)=(2^(n/2)).phi(((2^n).x(i)-k)*2^n+l); end aprox(x(i)*2^n+l)=aprox(x(i)*2"n+1)+...

theta(k-(Tl*2^n+2-2.N)+l).*phijk(i,k-(Ti.2^n+2-2.N)+1); end

end Z~ZZ~XXXX~X~Z~X~X~XXXXXX Z P l o t the r e s u l t s v e r s u s t he a n a l y t i c a l s o l u t i o n f i g u r e ( I )

plot(x,exact_x(l:length(x)),'-*',x,aprox,'v-') h = legend('Solution','Euler Aprox.',2); xlabel('0 \leq x \leq 2') title(' ') Z Display the approximation error disp(norm(exact_x(l:length(x))-aprox))

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